Vatican on a Wednesday

I've thought about doing this several ways, and nothing is perfect with regard to avoiding long queues and also midday heat, but to maximise time... Since the Colosseum opens a bit later than the Vatican (the park doesn't open until 9am), and because the Vatican visit is more structured, I would start the morning as early as possible at the Vatican. Do the museums tour, suffer the inescapable crowds, take a lunch break and then queue for the Basilica 30 minutes early (it is free). Take a serious break before you see the Colosseum in the later afternoon. I would probably not even bother with a ticket for the inside of the Colosseum (even RS, knowing it's a controversial opinion, says it's not critical). You have such limited time and the day is going to be exhausting anyway. You can take in the Forum and Palatine Hill at your own pace until 7pm which will be welcome after the Vatican crowds. Just my opinion. I haven't tried to do this myself, but I know both of these sites are brutally tiring in the hotter months but I also appreciate needing to take in as much as you can with such limited time.

Doing the museum tour early in the morning is not going to avoid the crowds: you won't actually be entering until 8:00 when doors open to the public, even though your meeting time (at a different location) is at 7:15. Still, it'll be cooler in there than in the afternoon. I'm assuming that you are visiting this summer/fall (high season)?

You can now buy and book a timed-entry ticket to St. Peter's through their official website that helps with the otherwise long queue:
https://www.basilicasanpietro.va/en/products/the-basilica

The Colosseum will be hot in the afternoon but your visit will likely be short, having worn your feet out at the Vatican. I wouldn't probably book more than the standard, timed-entry ticket if wanting to get a look at the interior: that's the "24h – Colosseum, Roman Forum and Palatine". You could visit the Forum and Palatine before or afterwards IF you're not completely knackered or too hot to trot by that point. As entry to the Palatine/Forum (they are a single attraction: passage between them is inside the security boundaries) isn't for a particular time, you could take a long-ish break between the Colosseum and those two sites as long as they're still open.

https://colosseo.it/en/opening-times-and-tickets/

I would absolutely do the Walks Pristine Sistine tour that you're referring to--done it multiple times and it's the one I recommend to my followers friends and family. As another poster said, it isn't going to avoid the crowds completely as it doesn't actually get you into the museum early, but it'll give you the most pleasant experience possible, in my opinion.

I was at the Papal Audience about a month ago. They estimate there are usually 60,000 people. Recommended arrival is 7am to get a good spot. Ended around 1130. Basically I am saying you don't want to be anywhere near the Vatican on a Wednesday morning. (PS: Papal Audiences are not a given. Depends on Pope's schedule).

...you don't want to be anywhere near the Vatican on a Wednesday
morning

Except that the entrance for the museums is just far enough from St. Peter's Square that I wouldn't think the audience crowds would be a real complication. At the time the OP would be entering the museums (8:00) the multitudes intent upon attending the audience would already be at the square or headed that direction.